Poles an Zeros

First, you have to wrap your mind around a plane that represents all possible frequencies in two dimensions, called the S plane.

The two dimensions of frequency are real and imaginary or time constant and sinusoidal. Time constants (decaying or growing) can be described as e (2.7 something, the base of natural logarithms) raised to a real power. Positive powers represent exponentially growing signals and negative exponents represent exponentially decreasing signals. The amount that any signal is growing or shrinking can be represented as a line across the S plane that is positioned along the real axis. If you make the exponent imaginary (has a square root of -1 factor in it) the result is a sine wave. Positive and negative imaginary exponentials are just sine waves going forward or backward in time (and they look exactly the same, since a sine wave is the same for all time). Imaginary frequencies are represented as a line across the S plane at right angles to the real exponential signals. Low frequencies are represented as being near the origin and higher frequencies are represented as being further from the origin. A zero frequency, representing DC, is the origin. So any exponentially growing or decaying sine wave of any frequency and decay or growth rate can be represented as a point on the S plane. Whew!

The amplitude of the signal of any frequency is represented as a distance at right angles to the S plane, like a rod sticking up out of it, for big values, or holes down into it for very small values. Actual linear systems frequency response to all frequencies can be represented as a surface or rubber sheet that is lifted up of held down at frequency points that represent important (resonant or characteristic) frequencies for the system. If the system produces some combination of sine wave and decay or growth rate if energized and left otherwise un-driven, (zero continuous input energy resulting in a finite result energy), that natural response can be said to be an infinite response (because it happens with zero input energy). Such points on the S plane are represented as having the rubber response surface held up at that point by an infinitely tall "pole". There may also be particular combinations of sinusoid and decay rate or growth that cannot be produced, regardless of how hard you drive the system. Those are "zero response points on th S plane rubber surface, and they can be thought of points tacked down to infinitely deep holes at those points on the rubber response surface. If the "rubber" surface is given the right kind of elasticity, mathematically, once a system is described at its "poles" and "zeros", the rubber surface predicts its response to all combinations of wave and decay/growth frequency by the height of its surface.

Is that enough abstraction for you to work on visualizing for a while?

The S plane frequency representation is based completely on representing the signals and responses as complex exponentials (e raised to an exponent with real and imaginary parts, representing the two dimensions of the S plane. If you Google "S plane" and "exponential frequency" you should have more hits than you need.

The S plane is an extremely useful visualization tool for capturing every possible response of a linear system to excitation by almost any signal.

--
Regards,

John Popelish

This is too big a subject for a news group reply. Read what John said, and read this:

If you're really interested in digital control, check out my book:

You _will_ have to wrap your brain around some mathematics, but I tried to make it as accessible as I could, and keep the discussion rooted in the real world at all times.

Basically, you can express the behavior of a linear, continuous-time, time-invariant system with a mathematical construct called a "transfer function". The transfer function is a consequence of analyzing a system using the Laplace transform (do a web search, or check Wikipedia).

Most such transfer functions (at least those that you don't run away from, screaming) are rational ratios of polynomials in s, the Laplace domain 'frequency' variable. So, I may have a PID controller whose transfer function is

(s + 250)(s + 1000) H(s) = ------------------- s(s + 5000)

This guy has zeros at s = -250 and s = -1000, and poles at s = 0 and s =

-5000.

It turns out to be very easy to calculate the response of such a system to a continuous sine wave at a particular frequency -- in this case the poles of a system will tend to make the system gain go down as the frequency is increased, and the zeros will make the gain go up.

You can also use the transfer function concept in the digital domain. Here the system has to be linear and shift-invariant, the analysis method is called the 'z transform', and the 'frequency domain' variable is z (instead of s).

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html

"Rob

** I have spent countless hours reading about & watching films and documentary stories about of the air war that dominated WW2. However, I have never heard of any confrontation occurring between the Polish air force ( or indeed any Polish WW2 flyer) and a Japanese Zero.

Probably, had a "one on one" showdown occurred somewhere over South East Asia, the Zero would have defeated the Pole.

The Pole would then become attenuated at a huge number of dB per thousand feet - until the poor fellow reached the stop band. Forever.

..... Phil

How about that! Phil has a sense of humor! Who could have imagined? Lets all go to the pub and hoist a pint!

OK, quick simplified version:

If you have a box with an input and an output, with linear response, there exists a mathematical expression, a transfer function, that describes how it behaves. If you know the transfer function, then for any input signal you can predict the output.

There are many ways to express the transfer function, most of them mathematically messy. Engineers prefer the Laplace Transform, which expresses the transfer function as a polynomial, using the complex varible "S". You'll have to read up on the theory to understand what S really is.

But if you express a transfer function as a polynomial on S, and factor it out nicely, and sweep S, the polynomial has "zeroes" where the numerator hits zero, and "poles" where the denominator hits zero. With a little practice, one can eyeball the equation, spot the poles and zeroes, and guess the frequency response.

Take this circuit:

input---------------R--------+--------output | | | C | | | ground

which is a simple "single-pole" resistor-capacitor lowpass filter. If R = 1 ohm and C = 1 farad, it has a pole at radian frequency w = 1, namely at 0.16 Hertz.

The transfer function is

1 / (S+1)

and

Output = Input * 1 / (S+1)

so has a pole at S = -1, sort of. (S is actually a complex variable... dig into the theory for gory details.)

You estimate the frequency response by pretending that S is the input frequency, in radians/second. Ignore the sign!

For very small S, very low frequencies, 1/(S+1) = 1, so frequency response is flat, unity gain. At high frequencies, S is big, so

1/(S+1) is almost the same as 1/S, so the output is dropping off inverse with frequency.

If you graph the frequency response, it will look close to...

| | |

1 | _______________________ | \\ | \\ G | \\ A | \\ I | \\ N | \\ | \\ | \\ | \\ | etc forever | | | | | | | | | __________________________________________________________ radian freq 1

where the gain is 1.00 at low frequencies up until 0.16 Hz, where it starts to roll off as 1/f, namely at -6 dB per octave. Engineers casually say that this frequency plot "has a pole at omega = 1" because the Laplace polynomial really does.

(Because S is actually a complex number, the rolloff region has a 90 degree phase shift. A 90 degree phase lag is associated with a 6 dB/octave rolloff in simple networks like this one.)

Now if you add a small resistor in series with that cap, say 0.1 ohms, it adds a zero. That looks like...

| | |

1 | _______________________ | \\ | \\ | \\ | \\ | \\ | \\ | \\ | \\ | \\ 0.1| ---------------------------- | | | | | | | | | __________________________________________________________ 1 10

where the second break is at w=10, namely 1.6 Hz. Phase is 0 in the flats and 90 lag on the slopey part.

I find it helpful to start with a rough, practical explanation of stuff like this before I hit the books for the formal stuff.

John

Actually, it may have happened. When Poland fell many poles emigrated to free countries to fight for the Allies. I imagine that most of them stayed close to the continent, but there may have been some polish nationals in the American, British, or Australian airplanes over the Pacific.

But a polish pilot in a PLZ "fighter", going against a zero, would have looked an awful lot like a small bird going against a falcon -- it would be a short exchange.

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

There were certainly Poles in the Royal Air Force, including during the Battle of Britain- see

among others.

-- Only three people have ever understood the Schleswig-Holstein problem One's dead, one's gone mad, and I've forgotten.

Just to fill in another detail: if one sets up the known conditions for a network of electrical components, the result is a set of simple linear equations. Solving the simultaneous equations will (at worst) result in an expression which can be simplified to a ratio of two polynomials in S, and the factoring of those polynomials is in general going to result in the numerator having zeroes (the zeroes of the ratio) and the denominator having zeroes (the poles of the ratio). The poles and zeroes express ALL of the resulting expression except for a single scaling constant.

The handling of a messy expression then is simplified to the values of some key numbers (the poles and zeroes).

"Tim Wescott"

** You mean the Polish PZL 11C - right?

....... Phil

Yse. Uh, Yes.

Apparently it was a very good fighter for 1932, or even 1935 -- the poles just didn't get the newer version in production soon enough for 1939. They had seen the writing on the wall, but much too late to save themselves.

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

Thank you all.

I guess I need to step back a bit before going into poles and zeros. I guess it all starts with the transfer function. I don't know what anybody else thinks but if there would be some interest would some of the more knowledgeable people be able to do a little tutorial of a practical example? I have Tim's book and a few others. I like Tim's book and I need to start going through it again. The first time I started reading it things didn't start coming together. I am a practical and graphical person and need to touch, see and smell to put everything together. I am now starting to put things together but need to understand a lot more. Would it be possible to do a tutorial using MathCAD, Mathlab, or other tools to do a simple regulator design. For example a small micro design which closes the loop (PIC or AVR)? Maybe a design like an old classic LM7805? I can do the electronics and someone else can help with the analysis, someone else could help with the firmware, someone else could help with corrections, etc. I am not proposing something which could be used as a product but something which mimics an IC already on the market. Something which is practical so it can be built and tested but also one can see what it takes as a design start. Maybe we could also look at different ways to close the loop and then continue on a servo design (but only mimicking devices on the market already, this is only for education) What does everyone think? If it sounds good then maybe I could setup a website so files can be exchanged?

Thoughts, suggestions? Rob

You can do most everyday closed-loop design for analog circuits, PLLs, and servo loops, using just Bode plots, which are graphical and intuitive, and not bother with the LaPlace pole/zero polynomial math. I haven't done that since college.

I actually prefer the graphical approach as opposed to the "big math" all-your-eggs-in-one-equation thing. The problem with the math is that it's easy to make one mistake somewhere in a big equation, and come up with truly silly results, and either miss the silliness or see it but not know why.

Bode based loop design is also better for estimating safety margins.

John

Hi John,

Thanks, I am not a big math person. I would like to be but I have a hard time making sense of all the nomenclature. It also seems to change depending on how one was taught. Even with doing it graphically a transfer function would still be required correct? So some big math would be required to put the transfer function in a form which can then be graphed. Or am I making a to big of deal out of this? How do you do it?

Thanks, Rob

Get a piece of semilog paper. Vertical (linear) axis will be gain in dB. Horizontal (log) axis is frequency. Actually, after a bit of practice, you can do it freehand, on plain paper or a whiteboard. We do this all the time.

Pick each little part of a circuit and sketch its frequency response.

An RC lowpass is a horizontal line of gain=1, that breaks at some corner frequency (where Xc = R) and then drops at -6 dB per octave, a straight 45 degree down slope on the graph.

0 dB ----------- \\ \\ -6 dB/octave \\ \\ \\ etc down forever

An opamp integrator is just a down slope, with a gain of 1 at the frequency where Xc = R.

\\ \\ -6 dB/octave \\ everywhere (except at \\ extremes, for the purists) \\ \\ \\

Simple RC networks can usually be graphed by inspection. Like...

in---------R1---------+--------------out | | R2 | | C | | gnd

usually plots like...

----------- \\ \\ \\ \\ \\ -----------------

So, for a closed-loop system, cascade all the gain curves around the loop; that's simple addition of the curves when you're working in dB's.

Recall that for simple networks like this, a 6 dB per octave down slope corresponds to 90 degrees phase lag.

The frequency/slope at which the overall closed loop crosses unity gain (0 dB) determines loop stability. If the slope is -6, phase margin is 90 degrees and the loop is heavily damped and bog stable. A little steeper, -9 db or around 45 degrees phase margin, is close to critically damped, snappier response. Steeper slopes, less phase margin, will make the loop start to ring. -12 dB or more is 180 degrees or more, unstable.

If you slide the whole curve up or down, you're simulating gain changes anywhere in the loop. Examine how that affects the slope of the zero crossing, and that tells you how much gain margin you have for stable operation.

With a little practice, you can do a lot of loop stabilization math in your head, and sketch the approximate Bode plot freehand, and get nice stable loops.

I could teach almost anybody how to do this in 20 minutes, if I had a whiteboard handy.

John